The minimum force needed to lift the weight will be the same as the force needed to hold the weight without falling.
So this is a statics question, find the force needed to keep everything where it is. Draw a free body diagram for the entire thing -- you should find the force F applied to the rope, the mass has some weight and there is some force applied by a shaft to hold the pulleys in place. Can you write an equation that will allow you to ignore the force applied by the shaft?